IIT JEE Blogs

Self adjusting Properties

Then x = sin θ                                                  …(2)

From (1) putting the value of θ in (2), we get,

Illustrations

Inverse of trigonometric ratios

We know that y = sin x means y is the value of sine of angle x if we consider domain and co-domain both as set R of real numbers. Sine ratio as seen from the fig. is many-one into function.

ü        But it is clear that if we restrict the domain to [-Π/2 , Π/2]  and range to [–1, 1], then. y = sin x is one-one onto and hence it is invertible.

So, y = sin x                             x ε [-Π/2 , Π/2] , y ε [–1, 1]

Þ  x = sin^–1 y                           y ε [–1, 1] , x ε [-Π/2 , Π/2]

ü        This value of x is called the principal value, i.e. belonging to [-Π/2 , Π/2]  and [-Π/2 , Π/2] range and it is called principal value range.

ü        The smallest numerical angle is called principal value.

ü        In general the inverse circular functions with their domain and range can be as given below:

Inverse Circular Function

Þ            sin^-1 x = θ        iff           sin θ = x, -Π/2 ≤ θ ≤ Π/2

Domain [-1,1]

Range [-Π/2 , Π/2]

Graph

Inverse Circular Function

Þ            cos^-1 x = θ   iff          cos θ = x, 0 ≤ θ ≤ Π

Domain [–1, 1]

Range [0, Π]

Graph

Inverse Circular Function

Þ            tan^-1 x = θ               iff   tan θ= x,  -Π/2< θ < Π/2

Domain (–∞, ∞)

Range (-Π/2 , Π/2)

Graph

• The mathematical definition of a function from set A to set B is that to each element a ε A there exists a unique element b ε B. As we know that in direct trigonometric functions, we are given the angle and we calculate the trigonometric ratio or the value at that angle. Also for many values of the angle, the values of trigonometric ratio is same. For example for sin Θ = 1/√2 , we have

Θ =Π/4 ,5Π/4 ,  9Π/4 etc. Now, direct trigonometric functions follow the definition of a function. But in inverse trigonometry, if we say that to a certain value of the trigonometric ratio there correspond many values of the angle, it violates the definition of function as it becomes a one – many relation. That is why, some restrictions have been imposed on the angles, and these are based on the principle values of the angle.

PRE-REQUISITE

ü        If A and B are two non–empty sets then a function from A to B associated to each element x in A, a unique element f(x) in B.

f : A → B

Þ            The set A is called Domain of f.

Þ            The set B is called the co-domain of f.

Þ      The range of f is the set consisting of all the images of the element of the domain A.

Þ      If          Range of f = {f(x) : x ε A}  then the function is onto.

Þ      One-One function: If x₁, x₂ ε A then f(x₁) = f(x₂) → x₁ = x₂

Þ            Contra positively x₁ ≠ x₂ ε f(x₁) ≠ f(x₂)

Þ      Many one function: f(x₁) = f(x₂)
where x₁ ≠ x₂.

Þ      Onto function: If f(A) = B i.e.
Range = co-domain then the function is onto.

Þ      Into function: If f(A) ς B the function is into

Þ      A function is invertible iff it is a one-one onto function. The inverse of a function is defined as if y = f(x),       x ε A, y ε B and f(x) is one-one and onto in A then            x = f ^–1(y) y ε B, x ε A

Þ      If A and B are domain and range of f(x) then B and A are those
of f^–1(x).